Abstract: New families of Cameron–Liebler line classes of \(\mathrm{PG}(3,q)\) , \(q\ge 7\) odd, with parameter \((q^2+1)/2\) are constructed. PubDate: 2019-03-01 DOI: 10.1007/s10801-018-0826-2

Abstract: As a homomorphic image of the hyperalgebra \(U_{q,R}(m n)\) associated with the quantum linear supergroup \(U_{\varvec{\upsilon }}(\mathfrak {gl}_{m n})\) , we first give a presentation for the q-Schur superalgebra \(S_{q,R}(m n,r)\) over a commutative ring R. We then develop a criterion for polynomial supermodules of \(U_{q,F}(m n)\) over a field F and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible \(S_{q,F}(m n,r)\) -supermodules for all r. As an application when \(m=n\ge r\) and motivated by the beautiful work (Brundan and Kujawa in J Algebraic Combin 18:13–39, 2003) in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra \(H_{q^2,F}({{\mathfrak {S}}}_r)\) ; see Brundan (Proc Lond Math Soc 77:551–581, 1998) for a proof without using the super theory. PubDate: 2019-02-02 DOI: 10.1007/s10801-019-00872-z

Abstract: We introduce an invariant of a finite point configuration \(A \subset \mathbb {R}^{1+n}\) which we denote the cuspidal form of A. We use this invariant to extend Esterov’s characterization of dual-defective point configurations to exponential sums; the dual variety associated with A has codimension at least 2 if and only if A does not contain any iterated circuit. PubDate: 2019-02-01 DOI: 10.1007/s10801-018-0816-4

Abstract: We study the relationship between depth and regularity of a homogeneous ideal I and those of (I, f) and I : f, where f is a linear form or a monomial. Our results have several interesting consequences on depth and regularity of edge ideals of hypergraphs and of powers of ideals. PubDate: 2019-02-01 DOI: 10.1007/s10801-018-0811-9

Abstract: The power graphP(G) of a group G is the graph whose vertex set is G, with x and y joined if one is a power of the other; the directed power graph \(\overrightarrow{P}(G)\) has the same vertex set, with an arc from x to y if y is a power of x. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most 2, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as \(\mathbb {Z}\) and \(\mathbb {Q}\) , we obtain more precise results. Any isomorphism \(P(\mathbb {Z})\rightarrow P(G)\) preserves orientation, so induces an isomorphism between directed power graphs; in the case of \(\mathbb {Q}\) , the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of \(\mathbb {Q}\) and about \(\mathbb {Q}^n\) . PubDate: 2019-02-01 DOI: 10.1007/s10801-018-0819-1

Abstract: Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart \(\delta \) -polynomials of these reflexive polytopes. PubDate: 2019-02-01 DOI: 10.1007/s10801-018-0817-3

Abstract: Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters \((v, m, k, \lambda )\) of a nontrivial SEDF that is near-complete (satisfying \(v=km+1\) ). We construct the first known nontrivial example of a \((v, m, k, \lambda )\) SEDF having \(m > 2\) . The parameters of this example are (243, 11, 22, 20), giving a near-complete SEDF, and its group is \(\mathbb {Z}_3^5\) . We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases \(m=2\) and \(m>2\) are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs. PubDate: 2019-02-01 DOI: 10.1007/s10801-018-0812-8

Abstract: A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p be a prime. It is known that there exist no tetravalent half-arc-transitive graphs of order p or 2p. Feng et al. (J Algebraic Combin 26:431–451, 2007) gave the classification of tetravalent half-arc-transitive graphs of order 4p. In this paper, a classification is given of all tetravalent half-arc-transitive graphs of order 8p. PubDate: 2019-01-31 DOI: 10.1007/s10801-019-00873-y

Abstract: In a recent paper, Cook et al. (Compos Math 154:2150–2194, 2018) used the splitting type of a line arrangement in the projective plane to study the number of conditions imposed by a general fat point of multiplicity j on the linear system of curves of degree \(j+1\) passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, they said that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property. PubDate: 2019-01-23 DOI: 10.1007/s10801-019-00871-0

Abstract: Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for Schur rings over cyclic groups, it is still a difficult problem to count all the partitions. For example, Kovacs, Liskovets, and Pöschel determine formulas to count the number of wreath-indecomposable Schur rings. In this paper, we solve the problem of counting the number of all Schur rings over cyclic groups of prime power order and draw some parallels with Higman’s PORC conjecture. PubDate: 2019-01-22 DOI: 10.1007/s10801-019-00870-1

Abstract: We present a method to derive the complete spectrum of the lift \(\varGamma ^\alpha \) of a base digraph \(\varGamma \) , with voltage assignment \(\alpha \) on a (finite) group G. The method is based on assigning to \(\varGamma \) a quotient-like matrix whose entries are elements of the group algebra \(\mathbb {C}[G]\) , which fully represents \(\varGamma ^{\alpha }\) . This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs. PubDate: 2019-01-02 DOI: 10.1007/s10801-018-0862-y

Abstract: In this paper, we prove that each matrix in \(M_{m\times n}({\mathbb {Z}}_{\ge 0})\) is uniformly column sign-coherent (Definition 2.2 (ii)) with respect to any \(n\times n\) skew-symmetrizable integer matrix (Corollary 3.3 (ii)). Using such matrices, we introduce the definition of irreducible skew-symmetrizable matrix (Definition 4.1). Based on this, the existence of maximal green sequences for skew-symmetrizable matrices is reduced to the existence of maximal green sequences for irreducible skew-symmetrizable matrices. PubDate: 2019-01-02 DOI: 10.1007/s10801-018-0861-z

Abstract: Let A be a K-subalgebra of the polynomial ring \(S=K[x_1,\ldots ,x_d]\) of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra \({\mathbb {G}}(A)\) of A is generated by monomials of degree \((r-1)d\) in S. We describe the generators and the structure of \({\mathbb {G}}(A)\) , when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of \({\mathbb {G}}(A)\) is bounded by the complexity of the graph G. PubDate: 2019-01-02 DOI: 10.1007/s10801-018-0865-8

Abstract: We estimate the number \( \mathcal {A}_{{\varvec{\lambda }}} \) of elements on a nonlinear family \(\mathcal {A}\) of monic polynomials of \(\mathbb {F}_{q}[T]\) of degree r having factorization pattern \({\varvec{\lambda }}:=1^{\lambda _1}2^{\lambda _2}\ldots r^{\lambda _r}\) . We show that \( \mathcal {A}_{{\varvec{\lambda }}} = \mathcal {T}({\varvec{\lambda }})\,q^{r-m}+\mathcal {O}(q^{r-m-{1}/{2}})\) , where \(\mathcal {T}({\varvec{\lambda }})\) is the proportion of elements of the symmetric group of r elements with cycle pattern \({\varvec{\lambda }}\) and m is the codimension of \(\mathcal {A}\) . We provide explicit upper bounds for the constants underlying the \(\mathcal {O}\) -notation in terms of \({\varvec{\lambda }}\) and \(\mathcal {A}\) with “good” behavior. We also apply these results to analyze the average-case complexity of the classical factorization algorithm restricted to \(\mathcal {A}\) , showing that it behaves as good as in the general case. PubDate: 2019-01-01 DOI: 10.1007/s10801-018-0869-4

Authors:Jiyong Chen; Wei Jin; Cai Heng Li Abstract: A complete classification is given of 2-distance-transitive circulants, which shows that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor. PubDate: 2018-05-22 DOI: 10.1007/s10801-018-0825-3

Authors:Giorgio Donati; Nicola Durante Abstract: Using a variation of Seydewitz’s method of projective generation of quadratic cones, we define an algebraic surface of \(\mathop {\mathrm{PG}}(3,q^n)\) , called \(\sigma \) -cone, whose \(\mathbb {F}_{q^n}\) -rational points are the union of a line with a set \(\mathcal{A}\) of \(q^{2n}\) points. If \(q^n=2^{2h+1}, h\ge 1\) , and \(\sigma \) is the automorphism of \(\mathbb {F}_{q^n}\) given by \(x \mapsto x^{2^h} \) , then the set \(\mathcal{A}\) is the affine set of the Lüneburg spread of \(\mathop {\mathrm{PG}}(3,q^n)\) . If \(n=2\) and \(\sigma \) is the involutory automorphism of \(\mathbb {F}_{q^2}\) , then a \(\sigma \) -cone is a subset of a Hermitian cone and the set \(\mathcal{A}\) is the union of q non-degenerate Hermitian curves. PubDate: 2018-04-23 DOI: 10.1007/s10801-018-0824-4

Authors:Jiangmin Pan; Zhaofei Peng Abstract: Edge-transitive graphs of order a prime or a product of two distinct primes with any positive integer valency, and of square-free order with valency at most 7 have been classified by a series of papers. In this paper, a complete classification is given of edge-transitive Cayley graphs of square-free order with valency less than the smallest prime divisor of the order. This leads to new constructions of infinite families of both arc-regular Cayley graphs and edge-regular Cayley graphs (so half-transitive). Also, as by-products, it is proved that, for any given positive integers \(k,s\ge 1\) and \(m,n\ge 2\) , there are infinitely many arc-regular normal circulants of valency 2k and order a product of s primes, and there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes; such arc-regular and edge-regular examples are also specifically constructed. PubDate: 2018-04-05 DOI: 10.1007/s10801-018-0823-5

Authors:Thomas Gerber Abstract: We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the Boson–Fermion correspondence and looking at the action of the Schur functions at \(q=0\) . In addition, we give the explicit formula for computing this crystal in full generality. PubDate: 2018-03-19 DOI: 10.1007/s10801-018-0820-8

Authors:Wenwen Fan; Cai Heng Li; Naer Wang Abstract: It is shown that a bipartite multi-graph \({\varGamma }\) has an orientably edge-transitive embedding with a single face if and only if \({\varGamma }=\mathbf{K}_{m,n}^{(\lambda )}\) such that \(\gcd (m,n)=1\) and mn is even whenever \(\lambda \) is even. A consequence of this shows that each orientable surface carries a simple edge-transitive map with a single face. PubDate: 2018-03-16 DOI: 10.1007/s10801-018-0821-7

Authors:Tao Zhang; Gennian Ge Abstract: Let C(d, k) and AC(d, k) be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree d and diameter k. It is well known that \(C(d,k)\le 1+d+d(d-1)+\cdots +d(d-1)^{k-1}\) with equality satisfied if and only if the graph is a Moore graph. However, there is a much better upper bound for abelian Cayley graph. We have \(AC(d,2)\le \frac{d^{2}}{2}+d+1\) and \(AC(d,k)\le \frac{d^{k}}{k!}+O(d^{k-1})\) . On the other hand, the best currently lower bounds are \(C(d,2)\ge 0.684d^{2}\) , \(AC(d,2)\ge \frac{25}{64}d^{2}-2.1d^{1.525}\) and \(AC(d,k)\ge (\frac{d}{k})^{k}+O(d^{k-1})\) for sufficiently large d. In this paper, we improve previous results on the degree–diameter problem. We show that \(C(d,2)\ge \frac{200}{289}d^{2}-5.4d^{1.525}\) , \(AC(d,2)\ge \frac{27}{64}d^{2}-3.9d^{1.525}\) and \(AC(d,k)\ge (\frac{3}{3k-1})^{k}d^{k}+O(d^{k-0.475})\) for sufficiently large d. PubDate: 2018-03-16 DOI: 10.1007/s10801-018-0822-6